Simplify the following expression and state the condition under which the simplification is valid. You can assume that $p \neq 0$. $k = \dfrac{-2}{30p + 80} \div \dfrac{p}{24p + 64} $
Dividing by an expression is the same as multiplying by its inverse. $k = \dfrac{-2}{30p + 80} \times \dfrac{24p + 64}{p} $ When multiplying fractions, we multiply the numerators and the denominators. $k = \dfrac{ -2 \times (24p + 64) } { (30p + 80) \times p } $ $ k = \dfrac {-2 \times 8(3p + 8)} {p \times 10(3p + 8)} $ $ k = \dfrac{-16(3p + 8)}{10p(3p + 8)} $ We can cancel the $3p + 8$ so long as $3p + 8 \neq 0$ Therefore $p \neq -\dfrac{8}{3}$ $k = \dfrac{-16 \cancel{(3p + 8})}{10p \cancel{(3p + 8)}} = -\dfrac{16}{10p} = -\dfrac{8}{5p} $